ISMT
Bayesian inference and time series analysis
Bayesian inference and time series analysis
Kartei Details
| Karten | 14 |
|---|---|
| Sprache | English |
| Kategorie | Mathematik |
| Stufe | Universität |
| Erstellt / Aktualisiert | 02.10.2022 / 05.02.2024 |
| Lizenzierung | Kein Urheberrechtsschutz (CC0) |
| Weblink |
https://card2brain.ch/box/20221002_ismte136
|
| Einbinden |
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[Time series analysis]
If AR(1) is causal and given as
\(x_t=\phi x_{t-1}+w_t\text{, where }w_t\sim wn(0, \sigma_w^2)\)
(a) Stationary solution?
(b) \(E(x_t)= \text{?}\)
(c) \(\gamma(h)= \text{?}\)
(d) \(\rho(h)= \text{?}\)
[Time series analysis]
If MA(1) is given as
\(x_t=\theta w_{t-1}+w_t\text{, where }w_t\sim wn(0, \sigma_w^2)\)
(a) \(E(x_t) = \text{?}\)
(b) \(\gamma(h)= \text{?}\)
(c) \(\rho(h)= \text{?}\)
[Time series]
In general, the correlation of any (stationary) time series can be calculated through...?
\(\rho(h)=\frac{\gamma(h)}{\gamma(0)}\)
Note:
- \(\gamma(0)\) is the variance of the series
- This only works due to stationarity. Pearson correlation coefficient is actually \(\frac{cov(X, Y)}{\sigma_X\sigma_Y}\)
[Statistics]
\(Var(aX) = \text{?}\)
\(Var(aX) = a^2Var(X)\)
Easy proof:
\(Var(aX)=Cov(aX,aX)=E[aXaX]-E[aX]E[aX]\)
\(=a^2E[X^2]-a^2E[X]E[X]=a^2\underbrace{\left[E[x^2]-E[X]E[X]\right]}_{Var(X)}\)
